56
ATTRACTIONS.
[ART. Ill
HO. Since V is regarded as a function of a, £, 7, we have dV _dV dV dV fa-da^dp^d^7*'
with similar expressions for dV/dy and dV/dz. These we differentiate again and substitute in Poisson's equation. Since the surfaces a, /S, 7 are orthogonal the coefficients of dzV/dadp <fec. are zero. We therefore have
Let the arbitrary functions a, p, y he so chosen that they satisfy Laplace's equation. The Poisson equation then becomes'
Let a be the potential of a thin ellipsoidal shell of unit mass, such as that described in Art. 68. Let (ale] be its serniaxes. It will be shown in the chapter on the attraction of ellipsoids that the level surfaces of the shell are the confocal ellipsoids. Let .(o/&V), (a" Ac,), (of" &c.) be the semiaxes of the three confocals which pass through any external point P.
Since 7?12=aa.2+a2/2 + a/, it is evident that \ is the component of force at P due to the shell in a direction normal to the ellipsoid (a'b'c'). It will also be shown
that this force is 7^= j^= --—-?> where p' is the perpendicular from the centre dp a o c
on the tangent plane. Similar expressions must hold for the hyperbolic confocals by the principle of continuity.
If Dlt D2, are the semi-diameters of the confocal ellipsoid respectively parallel to the normals at P to the confocal hyperboloids we know that p'D-J)^ a'b'c' by the properties of conjugate diameters. Also by the properties of confocal quadrics jD^ssa'3-^"2, D22=a'2-a'//2 and p'dp'=afdar. By using these expressions, we put the equation (1) into the form
a'"2)
,
(a"'2 - a") 2_
= ±*P (a."*- - a"") (a'« - a") (a* -
Since p'dp'=a'dd' the potentials a, /3, 7 are to be found from
____ _ ___ _ ___ __ __
dar~ b'c" da""" b"c"' da'"~~ b'"c'"' This form of Poisson's equation agrees with that given by Lame".
\
Theorems on the Potential.
\ ' ^ 111. The potential of any attracting system cannot be an absolute maximum or minimum at any point unoccupied by matter*.
If V be the value of the potential at any point P whose
* The theorems in this section may for the most part be found in Gauss' memoir on Forces varying inversely as the square of the distance, 1840. In the Cambridge and Dublin Mathematical Journal, Vol. iv. 1849, there is an interesting collection of theorems on the potential by Sir G. Stokes. Most of these were already known, but the proofs were much improved and put into new and better forms. This paper is reprinted in his collected works Vol. i. p. 104. The reader may also refer to papers by Lord Kelvin in various volumes of the Cambridge and Dublin Mathematical Journal, 1842 and 1843, reprinted in his Electricity and Magnetism. There is also a memoir by Chasles in the additions to the Connaissances des Temps for 1845.